To formalize this more general approach to prediction, define a probabilistic predictor as a map \(y^n \mapsto (\underline{\Pi }_{y^n}, \overline{\Pi }_{y^n})\)  that converts data into a pair of lower and upper predictive probabilities for \(Y_{n+1}\) . This includes the case of a precise predictive distribution \(\Pi _{y^n}\) , like that based on the Dirichlet process prior in the above example, if \(\underline{\Pi }_{y^n} = \overline{\Pi }_{y^n} = \Pi _{y^n}\) . We have opted for the “probabilistic predictor” terminology as opposed to “confidence predictor” in [1]} to highlight the difference between our objectives and theirs. As discussed above, we are interested in general uncertainty quantification about \(Y_{n+1}\) , beyond just prediction regions; for example, we anticipate evaluating “probabilities” for general assertions about \(Y_{n+1}\) . By an assertion about \(Y_{n+1}\) , we mean a subset of \(\mathbb {Y}\)  that may or may not contain \(Y_{n+1}\) . Note that assertions \(A\)  about \(Y_{n+1}\)  are virtually unlimited, e.g., \(A=\lbrace y \in \mathbb {Y}: \phi (y) \in B\rbrace \)  where \(\phi \)  is any relevant feature and \(B \subseteq \phi (\mathbb {Y})\)  is any relevant assertion about \(\phi (Y_{n+1})\) . Then the function \(A \mapsto (\underline{\Pi }_{Y^n}, \overline{\Pi }_{Y^n})(A)\)  provides uncertainty quantification about \(Y_{n+1} \in A\) , given data \(Y^n=y^n\) . To assess whether this uncertainty quantification is meaningful, below we consider properties of the sampling distribution of \((\underline{\Pi }_{Y^n}, \overline{\Pi }_{Y^n})(A)\)  as a function of the data \(Y^n\)  for fixed \(A\) .
